Some remarks on infinitesimals in MV-algebras

TL;DR

This paper explores the role of infinitesimals in MV-algebras, introducing quasiarchimedean and quasihyperarchimedean classes, and develops an algebraic geometry framework including a Nullstellensatz analogue.

Contribution

It characterizes new classes of MV-algebras related to infinitesimals and compact spectra, and provides elementary first-order proofs for their properties.

Findings

Characterization of quasihyperarchimedean MV-algebras.

Development of an algebraic geometry framework for MV-algebras.

Proof of a Nullstellensatz analogue for MV-algebras.

Abstract

Replacing $\{0\}$ by the whole ideal of infinitesimals yields a weaker notion of \emph{archimedean element} that we call \emph{quasiarchimedean}. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being \emph{quasihyperarchimedean} \mbox{MV-algebras,} which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the \mbox{set-theoretical} usually found in the literature.